- Thread starter Lazar
- Start date
- Tags instrumental variables

b =(X'X)^(-1)X'y

And since the instrumental variable variable (by wikipedia) is:

b_iv = (Z'X)^(-1)Z'y

(Where y is dependent variable, X the matrix of explanatory variables and Z the matrix of instrumental variables.)

And since in a generalized linear model (glm), in particular in a binary logit or probit model, maximum likelihood estimates is given by an iterative reweighed least squares (irls):

beta = (X'WX)^(-1) X'Wz

Here the W is a weight matrix (like in weighted least squares) and z is a pseudo-dependent variable (Taylor series of the link funktion) (Look at this)

Because of this I would guess and think that it is reasonable to use the following estimator, i.e. to just insert an instrumental variable in an irls.

beta_iv = (Z'WX)^(-1) Z'Wz

(The current beta_iv would give the current weight matrix W and that with the formula would give an updated value of beta_iv.)

I thought that one could just plug in this in R and use the matrix multiplication and get the result.

But Lazar asked for software in R.

When I looked a little bit more I saw this and under Instumental variables I saw the text "Binary responses : An IV probit model via GLS estimation is available in ivprobit ", and the packages ivprobit.

A related topic, Mendelian Randomization, is the coolest thing. This is when you have the same scenario, but you have a gene that randomizes you into a group and it is the upstream factor. E.g., lactose intolerance, alcohol dehydrogenase (poor processing of alcohol).